Let $\phi: \mathbb{R}^{n}\rightarrow \mathbb{R}$ be a regularizer combining lasso , group lasso, and fused lasso that is given by:
$\phi\left ( x \right ) = \lambda\left \| x \right \|_{1} + \mu\sum_{i=1}^{K}\left \| x_{G_{i}} \right \| + \gamma\sum_{j=1}^{n-1}\left | x_{j} - x_{j+1} \right |$
for some $\lambda,\mu,\gamma > 0$ where $G_{i} \subset \left \{ 1,2,...,n \right \}$ and $x_{G_{i}}$ is a subvector of $x$ indexed by $G_{i}$ for $i = 1,...,K$
I need to find the subdifferential $\partial \phi(x)$. For the first term, I know that the subdifferential $\partial (\lambda \|x\|_1)$ is the set of vectors $v$ such that $v_i \in \begin{cases} \{\lambda\} & x_i > 0 \\ \{-\lambda\} & x_i < 0 \\ [-\lambda, \lambda] & x_i = 0 \end{cases}$ but not sure on how to handle the rest of the terms. Any help would be much appreciated!I found this post subdifferential of fused lasso but not sure if that's correct